Incorporating “practical application” exercises into math courses gives students a rare chance to create their own math problems. The practice enables students to use higher order thinking skills and make connections between mathematics and fields of interest.
Since 1995, Northern Virginia Community College Math Professor Richard Semmler has incorporated "practical application" exercises into his calculus courses (Calculus with Analytic Geometry I and II). The original rationale behind this idea was threefold: 1) Give students the opportunity to connect the mathematics which they are learning to a field which they’re interested in. Mathematics is among the most abstract of disciplines; the immediate relevance of advanced mathematics such as calculus is generally not apparent to students, and advanced mathematics is usually taught in a decontextualized manner, i.e., without reference to real-world relevance. Having students create problems in fields they are interested in gives them the opportunity to establish a connection between the math they are learning and its real-world application. 2) Enable students to use higher-order thinking skills by creating math problems which apply what they’ve learned to an area of interest. Most math courses (including Semmler’s calculus courses) emphasize skill development through repeated problem solving. However, in mathematics education there is the danger of using problem solving exercises to teach "what to think" or "what to do” rather than "how to think." Mathematics textbooks often feature linear models which emphasize procedural knowledge of problem solving that are inconsistent with learning genuine problem solving (Wilson, Fernandez, and Hadaway 1993). “Problem-posing” is one method of promoting genuine problem solving by enabling students to use higher order thinking skills (specifically, Application in Bloom’s Taxonomy) than is possible from solving pre-fabricated problems. Having students practice creating new problems from old ones also helps students develop the skill of recognizing when to apply new technical skills toward solving problems (Education Development Center 2003). 3) Involve students more (inter)actively in online course activities. Every four weeks (four activities in total in the 16-week courses), students are required to design an example application and then solve it in its entirety. Semmler prefers that students select a problem related to their major area of interest (engineering, science, etc.) but says that "any project will suffice" so long as it pertains to the content studied in the related units. The application (problem + solution) is sent to the instructor via e-mail, who reviews it for accuracy. Students are allowed to use Scientific Notebook or other software to design this application. Semmler has observed that velocity and acceleration are quite popular topics in the first course and applications on exponential growth are quite popular in the second course. Assessment for each course consists of four exams with equal weight (25% of total grade). Assessment of the practical application activities is incorporated into the exam grade, with each activity worth 10 points on the corresponding exam. Semmler has incorporated practical application exercises into his Calculus with Analytic Geometry courses since 1995. In the first few years, selected student problems were posted on the computer conferencing system (FirstClass) for students in subsequent semesters to use as models for creating their own practical application exercises. Selected student-generated model problems are still made available to students in the current version of the course which uses the Blackboard LMS.
Over the years, students have submitted thousands of practical application exercises in the two courses. Semmler reports that most students submit relatively "run-of-the-mill" applications which are similar to textbook problems, but that "every now and then I will see a 'unique' application that is different from the usual" ones. Interestingly, Semmler also reports that "over the years I have not had any comments from students about the projects...Students have not 'complained' about doing the applications [either]...It is something they are expected to do once they are in the course and they complete the projects on a timely basis." These observations indicate that student-generated content can be easily and seamlessly integrated into math instruction. Semmler believes that student-generated practical application exercises are a "valuable tool" for learning calculus and analytic geometry topics. He notes that the exercises enable students "to look at a few applications from the book in preparation for designing and then solving their applications." This suggests that students benefit not only from using higher-order thinking skills (application, perhaps some analysis as well), but also from reviewing calculus problems from a producer perspective (i.e., with an eye toward creating a similar application) rather than a procedural one (i.e., just solve the problem). Semmler plans to keep this feature in his calculus courses "for years to come!!" Practical application exercises are essentially a no-cost method of providing opportunities for students to generate content as well as generating example problems for students to use as models in subsequent course offerings.
Wilson, Fernandez, and Hadaway (1993). Mathematical Problem Solving. Retrieved March 13, 2007 at: http://jwilson.coe.uga.edu/EMT725/PSsyn/PSsyn.html
Although a relatively modest example of student-generated content, it is still a significant step forward since most math students never get to create their own math problems.